You asked a simple question but the answer to "How does a CPU work?" can fill books. Yes modern processor use binary to represent numbers internally, and that is because it's easier to have and detect a voltage that is either high or low than various levels in between.
How binary numbers are added is no secret. The basic building block is a hardware logic circuit called a half adder. These are chained together to add any number of bits and to deal with the carries from the addition of lower order bits. Half adders and how they are chained together to make whole multi-bit adders is well covered out there, so I won't try to explain the details here. I think with the search terms "half adder" and "binary adder" you will easily find more than you ever wanted to know about them out there.
In practise, many adders are more complicated than a bunch of half adders chained together. This is soley to increase speed. Once you have looked up binary adder, you will see that information ripples from the low bit to the next bit to the next bit, etc. For wide addition, this takes time. There are various schemes to make this happen faster at the expense of more gates. The search term "lookahead carry" should yeild more than you want to know about this.
Hopefully this gets you started. Ask more specific questions if you want to know more after doing some reading.
Just to see what's out there I entered "half adder" and "binary adder" into Google. Lots of relevant-looking hits. Here is one that seems to explain it well enough. I'm sure there are others. This happened to be the first I clicked on, but it looks good enough: